For a given alphabetΣnormal-Σ\Sigma, the set of all languages over Σnormal-Σ\Sigma, as well as the set of all regular languages over Σnormal-Σ\Sigma, are examples of Kleene algebras. Similarly, sets of regular expressions (regular sets) over Σnormal-Σ\Sigma are a form (or close variant) of a Kleene algebra: let AAA be the set of all regular sets over a set Σnormal-Σ\Sigma of alphabets. Then AAA is a Kleene algebra if we identify ∅\varnothing as 000, the singleton containing the empty stringλλ\lambda as 111, concatenationoperation as ⋅normal-⋅\cdot, the union operation as ++, and the Kleene star operation as *{}^{*}. For example, let aaa be a set of regular expression, then